Defects and phase transitions to geometric phases of abelian GLSMs (Q6622497)

This paper considers two-dimensional gauged linear sigma models (GLSMs) with \(\left( 2,2\right) \)\ supersymmetry \(U\left( 1\right) \)\ gauge group, charged matter multiples and a superpotential, which can in general exhibit different phases dependently on the complexified FI parameter, characterized by a partial or total breaking of the gauge symmetry [\textit{E. Witten}, Nucl. Phys., B 403, No. 1--2, 159--222 (1993; Zbl 0910.14020); AMS/IP Stud. Adv. Math. 1, 143--211 (1997; Zbl 0910.14019)].\N\NThis paper aims to construct defect lines implementing the transition between different kinds of phases, more precisely from Landau-Ginzburg (LG) phases to geometric phases. The discussion is restricted to gauged linear sigma models with \(U\left( 1\right) \)\ gauge symmetry, exemplifying the general strategy. The authors expect that the arguments carry over to abelian gauge symmetries of higher rank in a straight forward way.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] briefly recalls how to describe B-type supersymmetirc boundary conditions and defects in the GLSM and its LG phases in terms of matrix factorization, particularly recalling the construction of in GLSMs given in [\textit{I. Brunner} et al., J. High Energy Phys. 2021, No. 5, Paper No. 6, 44 p. (2021; Zbl 1466.83116)]. \S 2.4 spells out the construction of transion defects between GLSMs and LG phases by pushing down the GLSM on one side of the identity defect to a LG phase. \S 2.5 explains how to push the GLSM on the other side of the resulting defect to the geometric phase, which shares features of matrix factorizations and complexes, being viewed as a complex of matrix factorizations or in terms of nested cones. \S 2.6 explains how to fuse the transition defects with boundary conditions, describing how D-branes behave under the transition.\N\N\item[\S 3] gives the concrete calculations for the specimen of the GLSM with superpotential\N\[\NW=PG\left( X_{1},\ldots,X_{N}\right)\N\]\Nwhere \(G\)\ is a homogeneous polynomial in the \(X_{i}\). In the geometric phase, these models reduce to nonlinear sigma models on the projective hypersurface\N\[\N\left\{ G\left( X_{1},\ldots,X_{N}\right) =0\right\} \subset \mathbb{P}^{N-1}\N\]\Nwhereas, at the LG point, they are effectively depicted by LG models with chiral superfields \(X_{i}\)\ and superpotential \(G\left( X_{1},\ldots ,X_{N}\right) \). Starting out with concrete formulas for the transition defects between GLSM and LG phase, it is shown that the associated functors map the D-branes for the LG phase to grade restricted subcategories of GLSM D-brane category in the sene of [\textit{M. Herbst} et al., ``Phases of $N=2$ theories In 1+1 dimensions with boundary'', Preprint, \url{arXiv:0803.2045}]. Subsequently, the authors construct the transition defects to the geometric phase, computing their action on D-branes. For the Calabi-Yau case, the procedure reproduces known result from [\textit{M. Herbst} et al., ``Phases of $N=2$ theories In 1+1 dimensions with boundary'', Preprint, \url{arXiv:0803.2045}] in a novel way.\N\N\item[\S 4] is a conclusion.\N\N\item[Appendix A] is concerned with reduction to finite rank.\N\end{itemize}